1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 441 We remark that, in the proof of the Sharafutdinov retraction Theorem ...

442 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 4. Integral curves to gradients of concave functions In this section we shall di ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 443 Note that by (H.32), the right tangent vector, if it exists, is unique. ...

444 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 4.2. Integral curves for concave functions and their properties. Recall that ctm ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 445 This implies</>' (s) = 1 a.e. Hence, by Lemma H.3 we have. (s) - ...

446 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS REMARK H.4 7. A nice example to keep in mind is the function given in Example H. ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 447 for s E (a, b). By Lemma H.15, for each s E (a, b) there exists a consta ...

448 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS COROLLARY H.50 (Uniqueness of integral curves for V' f /IV' fl^2 ). If x E C wit ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 449 PROOF. Assuming Lemma H.58 below on the existence of integral curves (wh ...

450 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS in particular, if IV f (Yo) I > c, then dc(y,ys) :S IVJ(:o)l-c. PROOF. By Lem ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 451 Hence, from the definition of y^8 , we have d (y, ys([V'f(yo)[-e1)) ::; ...

452 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS PROOF. Let C ~ df:ia~~. sup Since r (x, P) is a broken geodesic, where r (x, P)l ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 453 PROOF. (1) This is trivial. (2) This follows from Lemma H.55 directly. ( ...

454 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS PROOF. Note that the existence of partitions {Pk} is guaranteed by Corollary H.5 ...

INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 455 by (H.46). Applying Lemma H.37 to the concave function -f and by ap- ply ...

456 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS (ii) If f (x) < s::; f (y), then rs (y) = y, so that by Lemma H.52, de (rs (x ...

Appendix I. Asymptotic Cones and Sharafutdinov Retraction I keep on goin' guess I'll never know why. From "Life's Been Good" by ...

458 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION 1.1. Rays, Busemann functions and half-spaces. First we review a few notion ...

SHARAFUTDINOV RETRACTION THEOREM 459 The following property concerning approximating long geodesics by rays is useful (for a p ...

460 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION LEMMA I.8 (Sublevel set of bp)· For every choice of basepoint p E Mn and s ...